|
%I #14 Sep 29 2020 09:40:35
%S 1,2,3,6,12,26,59,138,332,814,2028,5118,13054,33598,87143,227542,
%T 597640,1577866,4185108,11146570,29798682,79932298,215072896,
%U 580327122,1569942098,4257254850,11569980794,31508150890,85968266198,234975421554
%N Antidiagonal sums of square table A086623.
%C a(n) is the number of Dyck (n+1)-paths (A000108) containing no DUDD and no UUPDD where P is a nonempty Dyck subpath. Example: a(2)=3 counts UUDDUD, UDUUDD, UDUDUD but omits UUUDDD because it contains an offending UUPDD and omits UUDUDD because it contains a DUDD. - _David Callan_, Oct 26 2006
%H Vincenzo Librandi, <a href="/A086625/b086625.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: A(x) = (1-x^2)/(1-x)^2 + x^2*A(x)^2.
%F a(n) ~ sqrt(11*r-3) / (4*sqrt(2*Pi)*(1-r)*n^(3/2)*r^(n+5/2)), where r = 0.3478103847799310287... is the root of the equation 4*r^3+4*r^2+r = 1. - _Vaclav Kotesovec_, Mar 22 2014
%F D-finite with recurrence (n+2)*a(n) +2*(-n-1)*a(n-1) +(-3*n+4)*a(n-2) +4*a(n-3) +4*(n-3)*a(n-4)=0. - _R. J. Mathar_, Sep 29 2020
%t CoefficientList[Series[(-1+x+Sqrt[1+x*(-2-3*x+4*x^3)])/(2*(-1+x)*x^2),{x,0,20}],x] (* _Vaclav Kotesovec_, Mar 22 2014 *)
%Y Cf. A086623 (table), A086624 (diagonal).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 24 2003
|
